Article pubs.acs.org/Macromolecules
Cite This: Macromolecules XXXX, XXX, XXX−XXX
Enhanced Segmental Dynamics of Poly(lactic acid) Glasses during Constant Strain Rate Deformation Trevor Bennin,* Josh Ricci, and M. D. Ediger Department of Chemistry, University of WisconsinMadison, Madison, Wisconsin 53706, United States
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S Supporting Information *
ABSTRACT: The combined effects of temperature and deformation on the segmental dynamics of poly(lactic acid) (PLA) glasses were investigated by using probe reorientation measurements. Constant strain rate deformations, with strain rates between 6 × 10−6 and 3 × 10−5 s−1, were performed on PLA glasses at temperatures between Tg − 15 K and Tg − 25 K. Deformation decreases the segmental relaxation time by up to a factor of 30 relative to the undeformed state. The segmental relaxation time in the postyield regime is related to the local strain rate via a power law, with exponents similar to those reported for lightly cross-linked PMMA. The Kohlrausch−Williams−Watts exponent, βKWW, commonly interpreted in terms of the width of the distribution of segmental relaxation times, changes from the undeformed state to the postyield regime, indicating a significant narrowing of the relaxation spectrum. We observe that βKWW is correlated to the deformation-induced increase of segmental mobility for PLA, as was reported for PMMA. The similar responses of PLA and PMMA to deformation suggest that the observed effects are the generic consequences of constant strain rate deformation on the segmental dynamics of polymer glasses.
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INTRODUCTION
complex physics of glasses and to greater predictive power for the manufacturing of advanced materials.6,10,11 Many theoretical approaches have been taken to describe the deformation of polymer glasses. As early as 1936, Eyring used a barrier-hopping model to describe the plasticity of solid materials.12 In his model, the stress imposed on a solid effectively decreased the height of barriers, allowing for faster molecular rearrangements and macroscopic flow. Models that employ a single relaxation time, like the fluidity model of Fielding, Larson, and Cates,13−15 the Eindhoven glassy polymer model,16 and Chen and Schweizer’s nonlinear Langevin equation (NLE) theory,17−19 have been used to describe nonlinear deformation even without explicitly including a distribution of relaxation times. Other models include, either implicitly or explicitly, a distribution of relaxation times to account for spatially heterogeneous dynamics. For example, the stochastic constitutive model (SCM) by Medvedev and Caruthers is a fully tensorial, mesoscopic model that has been used to describe physical aging and nonlinear deformation.20 It predicts a distribution of relaxation times which allows for deformation-induced changes in the dynamic heterogeneity to be described. Alternatively, the percolation of the free volume distribution (PFVD) model by Long and co-workers is also a mesoscopic model that incorporates a distribution of relaxation times. It is also able to describe physical aging21 and nonlinear deformation22 as well as changes in the dynamic heterogeneity of the system.23 These models rely on a variety of different mechanisms to
Polymer glasses are important engineering materials due to their many useful properties, such as their ease of processability, macroscopic homogeneity, and mechanical toughness. In common with other glassy materials, the linear mechanical properties of polymer glasses depend on the time scale over which they are measured: at short times, they act as elastic solids, while at long times, they can flow like viscous liquids.1,2 The relevant time scale for this change in properties of polymer glasses is the segmental relaxation time, which is typically longer than experimental time scales below the glass transition temperature, Tg. A crossover between elastic solidlike properties and viscous liquidlike properties can also be observed in the nonlinear deformation of polymer glasses. For example, during constant strain rate deformation, a polymer glass transitions from an initial elastic response into a postyield “flow” state, suggesting that the segmental relaxation time has accelerated in this process. In spite of decades of work, a fundamental description of the deformation of polymer glasses, in terms of the mechanical response and changes in segmental dynamics, has not been fully developed. This can be attributed to the nonlinear nature of the deformation process and the amorphous structure of glasses. In contrast to crystals, the local structure and dynamics of glasses are highly heterogeneous,3,4 which prevents the use of defect-based models commonly used for crystalline materials.5,6 Additional aspects of polymer deformation, such as strain hardening and the brittle−ductile transition, can also be attributed to the unique chain structure of polymers.7−9 An improved fundamental description of the deformation of polymer glasses will lead to a deeper understanding of the © XXXX American Chemical Society
Received: July 3, 2019 Revised: August 7, 2019
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DOI: 10.1021/acs.macromol.9b01363 Macromolecules XXXX, XXX, XXX−XXX
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for PLA and PMMA glasses in the plastic flow regime are the same. Finally, we find that the distribution of relaxation times for PLA narrows during deformation, similar to PMMA. These combined findings suggest that the observed effects are the generic response of polymer glasses undergoing constant strain rate deformation.
couple microscopic quantities (like density fluctuations, entropy fluctuations, and relaxation times) with macroscopic quantities (like stress and strain). While each model can effectively describe some aspects of the macroscopic mechanical response of a polymer glass to deformation, it is important to experimentally verify the underlying connection between the microscopic and macroscopic quantities. One important test for these models is to compare their predictions with measured changes in the segmental relaxation time of a glass in response to deformation. Many simulations and experiments have been performed to learn about the segmental dynamics of polymer glasses during deformation. Simulations of polymer glass deformation can directly measure accelerated dynamics in polymer glasses by calculating microscopic quantities that are typically unavailable in experiments, like particle hopping rates,24 bond autocorrelation functions,25,26 and intermediate scattering functions.24,25,27,28 Experimentally, mechanical testing29,30 has been used to infer changes in segmental dynamics during deformation.31 NMR,32 probe diffusion,33 and dielectric spectroscopy34,35 all provide direct evidence for enhanced segmental dynamics during deformation. For one system, lightly cross-linked PMMA glasses, probe reorientation techniques have identified and quantified the changes in the segmental dynamics during constant stress (creep)36−38 and constant strain rate deformations.39,40 In these probe reorientation experiments, it was shown that the relaxation time is strongly correlated to the strain rate in the plastic flow regime. It was also shown that deformation acts to narrow the distribution of segmental relaxation times, thus making the dynamics of a polymer glass more homogeneous. While the probe reorientation experiments have extensively characterized the segmental dynamics of lightly cross-linked PMMA during deformation, this system may not be ideal for identifying the generic response of polymer glasses due to the presence of cross-links41,42 and the existence of an overlapping secondary relaxation.43,44 Quantitative measurements of segmental dynamics during deformation of another polymer glass can help to determine the generality of the results of these previous experiments and provide additional quantitative data for the potential refinement of models of polymer glass deformation. Here we perform probe reorientation experiments on a poly(lactic acid) glass during constant strain rate tensile deformation at various temperatures somewhat below Tg. Strain rates between 6 × 10−6 and 3 × 10−5 s−1 were utilized at temperatures ranging from Tg − 25 K to Tg − 15 K. The segmental dynamics were monitored during deformation by using the probe reorientation technique. Parameters from the Kohlrausch−Williams−Watts function, τKWW and βKWW, representative of the average segmental relaxation time and the width of the distribution of relaxation times, respectively, were collected at various temperatures and strain rates, enabling a comparison to previous measurements on lightly cross-linked PMMA. We find that, at the coldest temperatures and fastest strain rates investigated, the segmental dynamics of PLA glasses can be accelerated by up to a factor of 30 during constant strain rate deformation. We show that the relaxation times of PLA glasses in the plastic flow regime are related to the local strain rate via a power-law relationship and that the change in the plastic flow relaxation time with strain rate for PLA is very similar to that of lightly cross-linked PMMA glasses. We also find that the temperature dependences of the relaxation times
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EXPERIMENTAL METHODS
Sample Preparation. Films of atactic poly(lactic acid) (PLA) were solvent-cast by using the following method. Poly(DL-lactide) pellets with Mn = 170 kg/mol, Đ = 1.05, and an L/D ratio of 50/50 were purchased (AP164, Akina, Inc.) and used without further purification. 0.8 g of PLA pellets was dissolved in 10 mL of dichloromethane in a glass vial. The fluorescent probe N,N′-dipentyl3,4,9,10-perylenedicarboximide (DPPC, Aldrich) was introduced to the solution to have a concentration between 10−7 and 10−6 M in the resulting polymer film. The structures of atactic PLA and DPPC are shown in Figure 1. The polymer/probe solution was filtered through a
Figure 1. Structures of the polymer and probe used in this work: atactic PLA (left) and DPPC (right). PTFE membrane with pore size 0.2 μm and transferred to flat-bottom glass dishes (diameter = 15 cm). The glass dishes were covered with aluminum foil in which a few small holes were punched to control the rate of evaporation. The solvent was evaporated over 24 h at room temperature. The films were removed from the glass dishes via sonication and placed in an oven at 335 K for 24 h to evaporate any remaining solvent. Samples were cut from the films using a custommade steel die that conforms to a scaled-down “dog-bone” shape of ASTM D1708-10. The cut samples had average thicknesses between 35 and 55 μm. The glass transition temperature of the material, Tg, was measured to be 325 ± 1 K as determined via DSC using the midpoint of the transition of the second heating scan at 10 K/min. Deformation Apparatus. The custom-built deformation apparatus utilized here has been previously described.39 The polymer sample is clamped into clips that are loaded into a temperature-controlled brass chamber. One clip is fixed in place while the other is attached to a poly(ether imide) rod that extends beyond the brass chamber and connects to a 5 N load cell to monitor the force applied to the sample. The load cell is attached to a U-shaped aluminum bar which is moved by a linear actuator. The position and velocity of the linear actuator control the extent and speed of the deformation. The bottom of the brass chamber is fitted with a glass window, and the entire deformation apparatus sits on top of a confocal microscope to allow for probe reorientation and local strain measurements during deformation. The constant global strain rate deformation was performed by controlling the linear actuator velocity. The deformation of glassy polymers can occur inhomogeneously, so a distinction must be made between the applied global strain rate and the local strain rate at the location of the optical measurements. The local strain rate is not controlled but is broadly targeted by applying an appropriate global strain rate; it is measured independently by taking images of bleached lines within the approximately 500 μm × 500 μm local measurement area. The local strain rate is calculated by using the change in the distance between the bleached lines between images. The local strain rate is typically equal to the global strain rate until yielding occurs, when strain localization occurs in the sample. After yield, the B
DOI: 10.1021/acs.macromol.9b01363 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules calculated local strain rate is effectively constant and larger than the applied global strain rate.39 The amount by which the local strain rate becomes larger than the global strain rate depends on the inhomogeneity of the deformation (i.e., the extent of strain localization) and the position of the optical experiment and can be up to a factor of 5. Representative plots of the local strain and global strain during deformation can be found in Figure S1. Thermal Protocol. The free-standing sample, clamped in the sample clips, is placed in an oven set to 335 K (Tg + 10 K) for 15 min to eliminate its thermal and mechanical history. Next, the sample is removed from the oven and cooled to room temperature (Tg − 30 K). The PLA sample is expected to rapidly cool through the glass transition, which has been reported to improve the ductility of PLA glasses.45 The sample is loaded into the deformation instrument 5 min after being removed from the oven. Once loaded, the temperature of the sample is increased at 1 K/min to the testing temperature, where the sample is then held isothermally for the remainder of the experiment. The sample is allowed to age for 30 min at the testing temperature before the deformation begins. The temperature accuracy of the apparatus is ±1 K, as determined by melting point tests. The temperature is stable to ±0.2 K during an experiment. Probe Reorientation Measurements. Probe reorientation measurements were performed during active deformation, as previously described.39,40 The sample is briefly exposed to a linearly polarized 532 nm laser beam which preferentially photobleaches probes with transition dipoles aligned with the polarization of the beam. Then a weak, circularly polarized beam induces fluorescence of the remaining unbleached probes. The fluorescence intensities parallel and perpendicular to the polarization of the bleaching beam are monitored and compared to intensities of a reference unbleached location to measure the anisotropy, r(t), as a function of time. The time decay of the anisotropy can be described with the Kohlrausch−Williams−Watts (KWW) function: r(t ) = r(0)e−(t / τKWW )
Figure 2. Comparison of anisotropy decay with and without large oscillations from strain-induced birefringence, controlled as explained in the text. The oscillating decay is captured by the sum of a stretched exponential and a sinusoid. The stretched exponential parameters for the oscillating and nonoscillating decays shown are log(τKWW/s) = 2.70, βKWW = 0.561 and log(τKWW/s) = 2.60, βKWW = 0.577, respectively. The inset is a schematic showing the position of photobleaching measurements (black pattern) and reference measurements that lead to anisotropy oscillations (1) and that do not lead to anisotropy oscillations (2).
tion-induced birefringence. Briefly, the changing thickness during deformation causes oscillations in the probe beam intensity inside the sample, leading to oscillations in the two orthogonally polarized fluorescence channels. The oscillations in the two fluorescence channels then become out-of-phase between the photobleached location and the reference location due to deformation-induced birefringence that varies along the deformation axis. We found that oscillations occurred in experiments where the reference intensities were displaced along the deformation axis from the photobleached location (box 1 in Figure 2 inset), while the oscillations were minimized when the reference intensities were measured in the same position along the deformation axis (box 2 in Figure 2 inset). Though we have seen oscillations in the polarized fluorescence intensities during previous measurements on PMMA during deformation, we have not seen oscillations in the anisotropy. Simple optical microscopy observations (not shown here) suggest that PLA has a larger birefringence than PMMA at the same postyield strain value, which is consistent with the magnitude of reported stress-optical coefficients of molten PLA and PMMA (3.1 and 0.15 GPa−1, respectively47−49). Thus, it is reasonable that anisotropy oscillations are observed in PLA but not PMMA. This explanation for the oscillations is also consistent with the observation that the frequency and amplitude of the fluorescence intensity oscillations are dependent on the local strain rate and the sample thickness, respectively. As a check on our understanding, we verified that the observed oscillations, in combination with a model of Fabry−Perot interference with corrections for a diverging light source, can accurately predict the local strain rate and thickness of the sample (not shown).
βKWW
Here, r(0) is the anisotropy immediately after the photobleaching event, τKWW is a characteristic segmental relaxation time, and βKWW describes the nonexponentiality of the decay, which is commonly interpreted in terms of the width of the distribution of relaxation times. Anisotropy data are fitted with a KWW function to extract τKWW and βKWW, which are presented in this work. Previous work has shown that DPPC, used here in PLA, is a good reporter of segmental dynamics in lightly cross-linked PMMA melts37 and glasses46 in the absence of deformation. Because the measurement of probe reorientation is not instantaneous, our probe reorientation times report an effective relaxation time that averages over a small range of strains. Measurements very near to the yield strain average over preyield and postyield mobility, giving a relaxation time longer than the plateau value (as seen in Figure 7). The plateau relaxation time is measured reproducibly once a probe reorientation measurement starts and ends in the postyield regime. While previous measurements of the anisotropy decay during the deformation of PMMA glasses showed a smooth decay, some measurements performed on PLA glasses during deformation showed a clear oscillatory behavior on top of the smooth decay. As explained below, we understand the origin of the oscillations, and they do not affect our measurement of the segmental dynamics of the system. Figure 2 shows two anisotropy measurements gathered one after the other in the postyield regime on the same sample; in this regime, the same segmental relaxation time is expected for both measurements. The oscillating anisotropy decay can be fit with the sum of a KWW function and a sinusoidal term as shown in Figure 2. Comparing the KWW parameters of the oscillating and smooth anisotropy decays shows that the oscillations do not affect our ability to extract an accurate τKWW and βKWW. The oscillatory anisotropy decay shown in Figure 2 is due to a combination of Fabry−Perot interference (induced by a change in thickness during deformation) and a position-dependent, deforma-
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RESULTS Segmental Dynamics of Quiescent PLA. Although the main focus of this work is the effect of constant strain rate deformation on the segmental dynamics of glassy PLA, it is important to establish that the probe reorientation technique can accurately describe the segmental dynamics of the quiescent system. Probe reorientation measurements were performed on quiescent PLA at various temperatures below Tg to generate anisotropy decays like those shown in Figure 3. C
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Figure 4. Relaxation times for PLA melts and glasses. The open squares are DMA data points extracted from Zuza et al.,53 shifted upward by 0.87 decades, and the filled circles are probe reorientation times measured in the present work at equilibrium, in the quiescent glass, and in the flow regime during isothermal deformation of the glass. The arrows represent the range of relaxation times measured during deformation while the pink line is representative of a single local strain rate, ε̇local = 10−4 s−1.
50/50, similar to the PLA studied here. The DMA data have been shifted vertically upward by 0.87 decades to align with the equilibrium probe reorientation measurements. Equilibrium relaxation times of supercooled liquids are commonly described by the empirical Vogel−Fulcher−Tammann (VFT) equation: B log(τ(T )/s) = A + T − T0
Figure 3. Anisotropy decays measured in quiescent PLA at various temperatures below Tg,DSC = 325 K. Filled circles are data points, and the solid lines are fits to a KWW function. Top: dynamics after isothermal aging to equilibrium. Bottom: dynamics after isothermal aging for 30 min.
The upper panel shows representative anisotropy decays for PLA that has been aged to equilibrium at temperatures between 315 and 323 K (Tg − 10 K to Tg − 2 K). These measurements were tracked over the course of 48 h and were considered to be in equilibrium when the anisotropy decay was found to be unchanging over the course of 24 h. The anisotropy for these equilibrium systems is well described by a KWW function with βKWW = 0.50. This is consistent with the range 0.46−0.53 from experiment50,51 and simulation52 as well as the value 0.49 obtained from our own analysis of dynamic mechanical data53 that displays the same dynamics as our system, as described below. The lower panel of Figure 3 shows anisotropy decays for glassy PLA between 300 and 310 K (Tg − 25 K to Tg − 15 K), temperatures also used for constant strain rate deformations. These data were gathered after isothermal aging for 30 min and are well described by a KWW function with βKWW = 0.42. The βKWW parameter for all KWW fits on quiescent PLA were constrained to 0.50 and 0.42 (for equilibrium data and aging data, respectively) to reduce the error in the estimate of the relaxation time. A plot similar to Figure 3 that includes the response for PLA at 317 and 315 K after isothermal aging for 30 min can be found in the Supporting Information (Figure S2). Figure 4 shows that the equilibrium relaxation times extracted from probe reorientation measurements in PLA (green data points) have the same temperature dependence as reported relaxation times of PLA measured from dynamic mechanical analysis;53 the PLA from ref 53 has a high molecular weight (Mv = 3.8 × 105 g/mol) and an L/D ratio of
Fitting the DMA data and the equilibrium probe reorientation times with the VFT equation yields A = −9.4, B = 421 K, and T0 = 283 K, shown by the green line in Figure 4. As expected, the relaxation times of the PLA glasses do not fall on the equilibrium line, but instead have a more Arrheniuslike temperature dependence, as shown by the blue data points in Figure 4. Also shown in Figure 4 are relaxation times measured during deformation, indicated by the pink data points and black arrows. We will return to these in the Discussion section. Segmental Dynamics of PLA Glasses during Deformation. The main results of this work focus on the segmental dynamics of glassy PLA during constant strain rate deformation. Typical stress−strain curves for PLA glasses deformed with a global strain rate ε̇ = 3.1 × 10−5 s−1 at different temperatures are shown in Figure 5. These curves show the expected features of deforming polymer glasses: at low strains (less than 0.01), the stress rises linearly with increasing strain; at higher strains, the stress begins to turn over and reaches a local maximum at the yield stress (the yield strain is 0.03); beyond yield, the stress slowly decreases and approaches the flow stress. In the experiments reported below, deformations were limited to global strains below 0.15 to minimize failure of the samples while allowing for sufficient time in the postyield regime to perform multiple probe reorientation measurements. Because of this, the strain hardening regime is not explored in these measurements. The elastic moduli for the deformations range between 1950 MPa at the lowest temperature and 1500 MPa at the highest D
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characterization of the decay rate and shape by the values of τKWW and βKWW, respectively, illuminates how constant strain rate deformation affects the segmental dynamics of PLA during deformation. Collections of anisotropy decays, like those shown in Figure 6, were obtained from glassy PLA samples held at various temperatures and deformed at various strain rates. Relaxation times extracted from these anisotropy decays are plotted as a function of strain in Figure 7 to show the evolution of the
Figure 5. Typical stress−strain curves for glassy PLA during constant strain rate tensile deformation.
temperature. These values fall within the range of values reported in the literature,45,54−56 though the method of preparation, thermal history, and stereochemical composition can modify the mechanical properties of PLA (with reported modulus values from 700 to 4000 MPa). Other features of the curves in Figure 5, like the increase of the yield stress and flow stress with decreasing temperature, are typical thermomechanical signatures of polymer glasses.2 Probe reorientation measurements were performed before and during deformation to track changes in the segmental dynamics. As seen by the changes in the anisotropy curves shown in Figure 6, segmental dynamics can be accelerated
Figure 7. Evolution of segmental relaxation time during constant strain rate deformation. The colored arrows represent the average relaxation time for the undeformed system at the specified temperature. The vertical black line (dashed) shows the yield strain. Local strain rates are averages in the postyield regime. Top: deformation at three different temperatures with the same local strain rate. Bottom: deformation at 300 K with four different local strain rates. Dashed curves are guides to the eye.
segmental dynamics during deformation. In both the upper and lower plots, the average relaxation time for the undeformed polymer at each temperature is indicated by colored arrows. The data are grouped and labeled by the average local strain rate, calculated as the average value of local strain rates measured in the postyield regime for each experiment in the group. The upper plot in Figure 7 shows how τKWW changes during deformations that occur at different temperatures, all with the same postyield local strain rate, ⟨ε̇local⟩ = 1.1 × 10−4 s−1. The lower plot shows the changes for τKWW during deformations at a single temperature (Tg − 25 K) with various postyield local strain rates. For all of the conditions shown, the segmental relaxation times decrease from the undeformed relaxation time until strains beyond the yield strain (represented by the black dashed line), when a plateau value is reached. A similar plot for the evolution of βKWW during deformation is shown in Figure
Figure 6. Typical normalized anisotropy decays gathered at increasing strain during constant strain rate deformation, indicating faster dynamics with increasing strain.
during a constant strain rate deformation. Figure 6 shows anisotropy curves collected at various local strains during a constant strain rate deformation at Tg − 20 K with an average postyield strain rate ⟨ε̇local⟩ = 1.1 × 10−4 s−1. Also shown are KWW fits to each anisotropy curve. The deformation conditions for this experiment caused the anisotropy to decay faster as the deformation proceeds, until strains well past the yield strain are reached. At this point, the anisotropy decays at the same rate, within experimental error. Quantitative E
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cause the same change in relaxation times as changing the temperature by 10 K. Together, these results show that the strain rate of the deformation is the dominant factor in determining the segmental dynamics. Though temperature still has an effect on segmental dynamics during deformation, its role is greatly reduced compared to the equilibrium melt and even to the quiescent glass. In the following discussion, we will compare the effects of constant strain rate deformation on the segmental dynamics of PLA glasses to those of lightly cross-linked PMMA glasses. In particular, we will show that the changes of the segmental relaxation times and the dynamic heterogeneity of these two polymer glasses are very similar. This argues that observed features represent the generic response of polymer glasses to constant strain rate deformation. Comparison of τKWW. The data in Figure 8 allow for a simple, direct comparison between the combined effects of temperature and strain rate on the postyield relaxation times in glasses of PLA and lightly cross-linked PMMA. This comparison is shown in the upper panel of Figure 9, which
S3. The change in the dynamics due to deformation depends on both the temperature and local strain rate; at a single local strain rate this change spans from a factor of 30 at Tg − 25 K to a factor of 4 at Tg − 15 K, while at a single temperature it spans from a factor of 30 at ⟨ε̇local⟩ = 9.5 × 10−5 s−1 to a factor of 5 at ⟨ε̇local⟩ = 1.5 × 10−5 s−1. To better understand the relationship between temperature, local strain rate, and segmental dynamics in glassy PLA, we will focus on measurements in the postyield regime where the segmental relaxation times are nearly constant. Figure 8 shows a strong correlation between the segmental relaxation time of glassy PLA in the postyield regime and the
Figure 8. Relaxation times versus local strain rate for measurements in the postyield regime. The dashed lines are power-law fits to the data. Slopes of the lines, from top to bottom, are −0.85 (±0.02), −0.82 (±0.04), and −0.77 (±0.02).
local strain rate for all temperatures investigated here. Each data point in Figure 8 represents a single probe reorientation time and local strain rate measurement. The strong correlation shown in Figure 8 is well-described by a power-law relationship, illustrated by the colored dashed lines. The slopes of the best-fit lines (with the standard error of the estimate) are −0.85 ± 0.02, −0.82 ± 0.04, and −0.77 ± 0.02 from top to bottom. This dependence of the relaxation time on the strain rate indicates that PLA glasses respond qualitatively like a shear-thinning fluid (with τKWW ∝ ηapparent ∝ ε̇n−1, n < 1) in the postyield regime of the deformation.
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DISCUSSION Figure 4 provides a useful overview of the results presented here on segmental dynamics in PLA glasses. The figure shows data collected for the quiescent glass (blue data points) and data collected for the glass during deformation at the highest strain rate in the postyield regime (pink data points). The relaxation times measured during deformation across the three temperatures explored here are much closer to each other than the relaxation times of the quiescent glasses across the same temperature range. Indeed, the temperature dependence of the dynamics during deformation is about 32 K/decade compared to about 9 K/decade for the dynamics of the quiescent glasses, both of which are much weaker than the observed temperature dependence of the equilibrium dynamics near Tg. On the other hand, the slopes described in Figure 7 indicate that changing the strain rate of a deformation by only a factor of 2.5 would
Figure 9. Top: relaxation times in the postyield regime versus local strain rate for PLA (filled circles and dashed lines) and PMMA (solid lines: fit to PMMA data from ref 40). PLA and PMMA relaxation times show very similar behavior in the postyield regime. Bottom: PLA data shifted downward by a factor of 3 to emphasize the similarity in the behavior.
combines the PLA data from Figure 8 with the solid lines showing fits to data on lightly cross-linked PMMA from ref 40. Upon comparison of the two polymers, the exponents of power-law fits (the slope of the lines) are the same (to within 0.1) at the same temperature relative to Tg, and for both polymers they appear to approach −1 as the temperature is decreased. The similarity of these responses is made more F
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Macromolecules apparent in the lower panel of Figure 9. Here the absolute values of τKWW for PLA glasses have be shifted downward by a factor of 3 to emphasize the similarity to the PMMA data for any given temperature relative to Tg. We emphasize that the power-law relationships shown in Figure 9 are effectively the same for the two polymers, despite the existence of an overlapping secondary relaxation in PMMA.43,44 The similarity of the exponents for the two systems as a function of temperature indicates that these are robust features of polymer glass deformation. The values for these exponents are near the value of −0.8 reported for colloidal glasses,57,58 and they are consistent with simulations of bead−spring polymers.26,27 NLE theory predicts similar values for these exponents in polymer glasses17 and colloidal glasses.59 The weak temperature dependence of the exponents can also be explained in the context of NLE theory.17 The lower panel of Figure 9 indicates that the absolute values of τKWW for PLA glasses are roughly 3 times longer than those observed for PMMA glasses for any given temperature and strain rate. The same factor of 3 has also been observed in the probe reorientation time scale in the two glasses measured during physical aging.46,60 In these experiments, the relaxation times of PLA60 and PMMA46 glasses are measured during physical aging through stress relaxation after a linear step strain deformation and with probe reorientation in the absence of deformation. When the two systems show the same mechanical relaxation time, the probe reorientation time scale for PLA is 0.5 decades longer than the probe reorientation time scale for PMMA, similar to the difference seen here. We attribute the difference in the probe reorientation time scale to the size of the probe relative to chain segments in each polymer system; i.e., the length scale of structural relaxation near Tg in PLA is smaller in absolute terms than the corresponding length scale for PMMA. In the absence of other interactions, it has been shown that increasing probe size for a given host polymer increases the probe reorientation time.61−63 Comparison of βKWW. Along with the characteristic relaxation time τKWW, the dynamic heterogeneity of PLA, indicated by the value of βKWW, is tracked during deformation. In Figure 10, the values of βKWW in the postyield regime obtained from PLA are compared to those obtained from PMMA in ref 40. βKWW for PLA increases from a value of 0.42 in the undeformed state to about 0.60 for the largest deformationinduced increases in segmental mobility, while βKWW for PMMA increases from 0.31 in the undeformed state to about 0.57. The increases in βKWW during the deformation of PMMA glasses have been interpreted previously as a narrowing of the distribution of relaxation times.37,39 Thus, for both PLA and PMMA, the more deformation decreases relaxation times (faster segmental dynamics), the more it narrows the original distribution of relaxation times. By representing the KWW function as a series of weighted exponentials,64 the distribution of relaxation times for specific βKWW can be constructed. This method indicates that the fwhm of the distribution of relaxation times for PLA decreases from 2.1 to 1.1 decades during deformation while that of PMMA decreases from 3.1 to 1.2 decades, a significant narrowing for both systems. While experiments have shown that cross-linking can affect the heterogeneity of the underlying dynamics in a PMMA glass,41 the changes in βKWW with deformation seen in both PLA and PMMA glasses suggest that a narrowing of the distribution of
Figure 10. Values of the β parameter from the KWW fit to anisotropy data collected in the postyield regime. The x-axis is a measure of the acceleration in dynamics compared to the initial undeformed system. Filled stars represent measurements on PLA, and open circles represent measurements on PMMA (from ref 40). Error bars represent one standard deviation. Dashed lines show linear fits through each data set.
relaxation times is a generic feature of polymer glass deformation. A narrowing of the distribution of relaxation times due to deformation is predicted by the stochastic constitutive model of Medvedev and Caruthers.20 The model incorporates material properties of a polymer glass to develop a fully tensorial, mesoscale description of the macroscopic response to deformation. An important feature of the model is the natural occurrence of a distribution of relaxation times, which is also seen to evolve during deformation. In particular, the model predicts that the distribution of relaxation times for a lightly cross-linked PMMA glass shifts to faster times and narrows as deformation proceeds into the flow regime during uniaxial tensile deformation.65 In ref 65, the fwhm of the distribution narrows from 4 to 2.4 decades during deformation with a strain rate ε̇ = 10−4 s−1. The change in fwhm agrees reasonably with the narrowing in PMMA described above. Furthermore, we fit the relaxation time distributions to a KWW distribution to extract τKWW and βKWW, which we then analyzed to determine the relationship between βKWW and the acceleration from the undeformed system. This analysis (not shown here) of PMMA results from ref 65 shows an average increase of 0.10 in βKWW for each decade of acceleration in the average relaxation time, consistent with the value of 0.12 increase per decade observed experimentally for PMMA. The agreement shown between the model and experiments for PMMA supports the mechanism that causes narrowing in the model. The new data for PLA presented here could be used to further test this mechanism. A coarse-grained model by Conca et al.23 has qualitatively reproduced changes in segmental dynamics of a polymer glass during constant strain rate deformation. This model relies on a facilitation mechanism to allow fast and slow mesoscopic subunits to relax during deformation. It also includes a distribution of relaxation times which evolves as the mesoscopic subunits relax. Full knowledge of the distribution of relaxation times at any given time during deformation allows for the authors to create a macroscopic relaxation function which can then be fit to a KWW function to mimic the analysis G
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done on probe reorientation measurements. The βKWW parameter from the model shows a qualitative similarity to experiments on lightly cross-linked PMMA, with βKWW increasing from the undeformed state to the plastic flow regime. As with the SCM above, we have analyzed the data provided in ref 23 to determine the relationship between βKWW and acceleration from the undeformed system. This analysis shows an average increase of 0.24 in βKWW for each decade of acceleration, nearly double what is observed experimentally in PMMA. It is possible that choosing alternate model parameters will allow for a more quantitative description of the experimental results for PMMA and also allow a description of the experimental results from PLA displayed here.
ACKNOWLEDGMENTS We thank the National Science Foundation (DMR-1708248) for support of this work. We thank Enran Xing for helpful discussions.
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CONCLUSION Here we have shown that the segmental dynamics of poly(lactic acid) glasses and lightly cross-linked poly(methyl methacrylate) glasses show the same generic behavior during constant strain rate deformation, as measured using the probe reorientation technique. The changes in the characteristic relaxation time of PLA glasses during plastic flow were shown to strongly depend on the local strain rate via a power-law relationship, with a weakly temperature-dependent exponent, which parallels previous results from lightly cross-linked PMMA. The temperature dependence of the relaxation time in the plastic flow regime was also shown to be the same for PLA and PMMA glasses. The distribution of relaxation times of PLA glasses was also shown to narrow during deformation, again like PMMA. The similarities reported here in the two polymer glass systems suggest that these effects are the generic consequences of constant strain rate deformation on the segmental dynamics of polymer glasses. Though some models can qualitatively describe the segmental dynamics of PMMA during constant strain rate deformation, this new work is anticipated to allow for further refinement of these models for a closer quantitative description. This would allow for a more complete understanding of the microscopic details of the deformation of polymer glasses and, in turn, better prediction of the mechanical and engineering properties of polymer glasses. ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.9b01363. Evolution of local strain during deformation; additional anisotropy decay measurements for quiescent PLA; βKWW during deformation (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail
[email protected]. ORCID
Trevor Bennin: 0000-0001-8509-6344 Josh Ricci: 0000-0003-4371-9093 M. D. Ediger: 0000-0003-4715-8473 Notes
The authors declare no competing financial interest. H
DOI: 10.1021/acs.macromol.9b01363 Macromolecules XXXX, XXX, XXX−XXX
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